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Timothy J. Healey
Professor and Director of Undergraduate Studies
Departments/Programs
 Mathematics
Graduate Fields
 Applied Mathematics
 Mathematics
 Theoretical and Applied Mechanics
Research
Applied analysis and partial differential equations, mathematical continuum mechanics
I work at the interface between the nonlinear analysis of PDE and the mechanics of elastic structures and materials. Although not as wellknown or popular as mathematical fluid mechanics, nonlinear finitedeformation elasticity is the central model of continuum solid mechanics. While the mathematical theory of linear(ized) elasticity is a classical, wellhoned subject, properly formulated problems of the nonlinear theory lead to open mathematical questions. For example, a systematic approach to the construction of weak solutions is still not known to this day.
The two main goals of my work are to establish rigorous existence results and to uncover new phenomena. Along with my coworkers and students, we develop and employ topologicaldegreetheoretic methods, direct methods of the calculus of variations, and global symmetrybreaking bifurcation theory to establish existence theorems. Examples illustrating the latter goal include things like wrinkling behavior in thin sheets and icosahedral pattern formation in 2phase lipid bilayer structures. The work involves a symbiotic interplay between three key ingredients: careful mechanicsbased modeling, mathematical analysis, and efficient computation.
Courses
Spring 2020
 MATH 4280 : Introduction to Partial Differential Equations
 MATH 4900 : Supervised Research
 MATH 4901 : Supervised Reading
 MATH 6160 : Partial Differential Equations
Fall 2020
Publications

Classical solutions in the large in incompressible nonlinear elasticity, Arch. Rational Mech. Anal. (2018) DOI 10.1007/s0020501801342y.

Direct construction of symmetrybreaking directions in bifurcation problems with spherical symmetry (with S. Dharmavaram), Discrete and Cont. Dynamical Systems S, 12 (2019) 16691684.

The Mullin effect in the wrinkling behavior of highly stretched thin films (with E. Fejér and A. Sipos), J. Mech. Phys. Solids 119 (2018) 417427.

Injectivity and selfcontact in secondgradient nonlinear elasticity (with A. Palmer), Calc. Var. PDE 56 (2017) no. 114, DOI 10.1007/s005260171212y.
 SymmetryBreaking Global Bifurcation in a Surface Continuum PhaseField Model for Lipid Bilayer Vesicles (with S. Dharmavaram), SIAM J. Math. Anal. 49 (2017) 1027–1059.
 Direct computation of twophase icosahedral equilibria of lipid bilayer vesicles (with Q. Li and S. Zhao), Comput. Methods Appl. Mech. Engrg. 314 (2017) 164–179.

Stability boundaries for wrinkling in highly stretched elastic sheets (with Q. Li), J. Mech. Phys. Solids 97, (2016) 260274.
 Injective weak solutions in secondgradient nonlinear elasticity (with S. Krömer), ESAIM: COCV 15 (2009) 863–871.
 Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids 7 (2002) 405–420.
 Global continuation in nonlinear elasticity (with H. Simpson), Arch. Rational Mech. Anal. 143 (1998) 1–28.
 Preservation of nodal structure on global bifurcating solution branches of elliptic equations with symmetry (with H. Kielhöfer), JDE 106 (1993) 7089